MATH
Sec_2.4

# Sec_2.4 - 2.4 Exact Equations Theorem 2.2 Criterion for an...

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2.4 Exact Equations Theorem 2.2 Criterion for an Exact Differential Let ( 29 , M x y and ( 29 , N x y be continuous and have continuous first partial derivatives in a rectangular region R defined by a x b < < and c y d < < . Then a necessary and sufficient condition that ( 29 ( 29 , , M x y dx N x y dy + be an exact differential is M N y x x = . Ex. Show that the equation is exact. a. ( 29 ( 29 2 2 5 5 3 0 x y dx x y dy - + - + = ( 29 , 2 5 M x y x y = - and ( 29 2 , 5 3 N x y x y = - + . Then ( 29 ( 29 2 5 3 2 5 5, 5 x y x y y x x - + x - = - = - . It is an exact equation. Let’s look at the equation if 2 3 5 x xy y c - + = , then ( 29 2 3 5 2 5 x xy y x y x x - + = - and ( 29 2 3 2 5 5 3 x xy y x y y x - + = - + . Clearly, If the function ( 29 , f x y c = is a solution to the exact differential equation, then ( 29 ( 29 ( 29 ( 29 , , , , f x y f x y M x y and N x y x y x = = . Method of Proof & Solution ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , , , , , f x y M x y f x y M x y dx g y x f M x y dx g y N x y y y x = = + ° ° ° = + = ° ° °

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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , 0 , , 0 , 0 0. g y N x y M x y dx y g y x N x y M x y dx x y N M x y dx x y x N M x y x x = - ° ° ° = ° ° - = °
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